This book provides a comprehensive introduction to the systematic theory of tensor products and tensor norms within the framework of operator spaces. The use of tensor products has significantly advanced functional analysis and other areas of mathematics and physics, and the field of operator spaces is no exception. Building on the theory of tensor products in Banach spaces, this work adapts the definitions and results to the operator space context. This approach goes beyond a mere translation of existing results. It introduces new insights, techniques, and hypotheses to address the many challenges of the non-commutative setting, revealing several notable differences to the classical theory. This text is expected to be a valuable resource for researchers and advanced students in functional analysis, operator theory, and related fields, offering new perspectives for both the mathematics and physics communities. By presenting several open problems, it also serves as a potential source for further research, particularly for those working in operator spaces or operator algebras.

We introduce a notion of completely bounded holomorphic functions defined on the open unit ball of an operator space. We endow the set of these functions with an operator space structure, and in the scalar-valued case we identify an operator space predual for it which is a noncommutative version of Mujica's predual for the space of bounded holomorphic functions and satisfies similar properties. In particular, our predual is a free holomorphic operator space in the sense that it satisfies a linearization property for vector-valued completely bounded holomorphic functions. Additionally, several different operator space approximation properties transfer between the predual and the domain.

We study a notion analogous to the \(p\)-Approximation Property (\(p\)-AP) for Banach spaces, within the noncommutative context of operator spaces. Referred to as the \(p\)-Operator Approximation Property (\(p\)-OAP), this concept is linked to the ideal of operator \(p\)-compact mappings. We present several equivalent characterizations based on the density of finite-rank mappings within specific spaces for different topologies, and also one in terms of a slice mapping property. Additionally, we investigate how this property transfers from the dual or bidual to the original space. As an application, the \(p\)-OAP for the reduced \(C^*\)-algebra of a discrete group implies that operator \(p\)-compact Herz-Schur multipliers can be approximated in completely bounded norm by finitely supported multipliers.

We continue our study of the mapping ideal of operator \(p\)-compact maps, previously introduced by the authors. Our approach embraces a more geometric perspective, delving into the interplay between operator \(p\)-compact mappings and matrix sets, specifically we provide a quantitative notion of operator \(p\)-compactness for the latter. In particular, we consider operator \(p\)-compactness in the bidual and its relation with this property in the original space. Also, we deepen our understanding of the connections between these mapping ideals and other significant ones (e.g., completely \(p\)-summing, completely \(p\)-nuclear).

We initiate the study of the small scale geometry of operator spaces. The authors have previously shown that a map between operator spaces which is completely coarse (that is, the sequence of its amplifications is equi-coarse) must be \(\mathbb{R}\)-linear. We obtain a generalization of the aforementioned result to completely coarse maps defined on the unit ball of an operator space. By relaxing the condition to a small scale one, we prove that there are many non-linear examples of maps which are completely Lipshitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps.

The classical Mazur map is a uniform homeomorphism between the unit spheres of \(L_p\) spaces, and the version for noncommutative \(L_p\) spaces has the same property. Odell and Schlumprecht [The distortion problem, Acta Math.173 (1994), no. 2, 259–281] used two types of generalized Mazur maps to prove that the unit sphere of a Banach space \(X\) with an unconditional basis is uniformly homeomorphic to the unit sphere of a Hilbert space if and only if \(X\) does not contain \(\ell_\infty^n\)'s uniformly. We prove a noncommutative version of this result, yielding uniform homeomorphisms between spheres of unitarily invariant ideals, and along the way we study noncommutative versions of the aforementioned generalized Mazur maps: one based on the \(p\)-convexification procedure, and one based on the minimization of quantum relative entropy. The main result provides new examples of Banach spaces whose unit spheres are uniformly homeomorphic to the unit sphere of a Hilbert space (in fact, spaces with the property (H) of Kasparov and Yu).

We show that there is an operator space notion of Lipschitz embeddability between operator spaces which is strictly weaker than its linear counterpart but which is still strong enough to impose linear restrictions on operator space structures. This shows that there is a nontrivial theory of nonlinear geometry for operator spaces and it answers a question in [B. M. Braga and J. A. Chávez-Domínguez, Completely coarse maps are \(\mathbb{R}\)-linear, Proc. Amer. Math. Soc. 149 (2021), no. 3, 1139–1149]. For that, we introduce the operator space version of Lipschitz-free Banach spaces and prove several properties of it. In particular, we show that separable operator spaces satisfy a sort of isometric Lipschitz-lifting property in the sense of G. Godefroy and N. Kalton. Gateaux differentiability of Lipschitz maps in the operator space category is also studied.

Braga and Sinclair [A Ribe theorem for noncommutative \(L_1\)-spaces and Lipschitz free operator spaces, arXiv:1911.05505] have recently introduced the notion of completely coarse map, namely, a map between operator spaces such that the family of its amplifications is equi-coarse. In this paper we prove that such a completely coarse map is in fact real-linear, answering multiple questions from Braga and Sinclair; the proof relies on the existence of Hadamard matrices of arbitrarily large sizes. We then introduce a notion of nonlinear embeddability between operator spaces which is strictly weaker, but which is still strong enough so that it preserves some features of the operator space structure.

In the present paper, we introduce a new concept of positive \(p\)-majorizing operators as a dual notion of positive \(p\)-summing operators and generalize the concept of majorizing operators introduced by Schaefer [Isr J Math 13:400–415, 1972]. We introduce the concept of positive \((p, q)\)-dominated operators and prove a positive version of the famous Kwapień’s factorization theorem for \((p, q)\)-dominated operators via positive \(p\)-majorizing operators. We also introduce the notion of disjoint \(p\)-summing operators which is a new larger class of operators than positive \(p\)-summing operators and use it to characterize the Radon–Nikodým property. Finally, we investigate the maximal properties of these four classes of operators and prove that they are maximal in corresponding sense.

We generalize the notions of asymptotic dimension and coarse embeddings from metric spaces to quantum metric spaces in the sense of Kuperberg and Weaver. We show that quantum asymptotic dimension behaves well with respect to metric quotients and direct sums, and is preserved under quantum coarse embeddings. Moreover, we prove that a quantum metric space that equi-coarsely contains a sequence of expanders must have infinite asymptotic dimension. This is done by proving a quantum version of a vertex-isoperimetric inequality for expanders, based upon a previously known edge-isoperimetric one due to Temme, Kastoryano, Ruskai, Wolf, and Verstraete.

In Quantum Information Theory there is a construction for quantum channels, appropriately called a quantum graph, that generalizes the confusability graph construction for classical channels in classical information theory. In this paper a definition of connectedness for quantum graphs is provided, which generalizes the classical definition. It is shown that several examples of well-known quantum graphs (quantum Hamming cubes and quantum expanders) are connected. A quantum version of a particular case of the classical tree-packing theorem from Graph Theory is also proved. Generalizations for the related notions of k-connectedness and of orthogonal representation are also proposed for quantum graphs, and it is shown that orthogonal representations have the same implications for connectedness as they do in the classical case.

We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev ones on such graphs. As is often the case when relating isoperimetric and Sobolev inequalities, our crucial tool is a coarea formula; we obtain it as a variation of one previously proved by Lange, Liu, Peyerimhoff and Post, [Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 4165–4196]. As a first application, we show that the signed Cheeger constant behaves additively with respect to Cartesian products of graphs. Using heat kernel techniques, we also give lower bounds for the eigenvalues of the discrete magnetic Laplacian.

The Ando-Choi-Effros lifting theorem provides conditions under which a bounded linear mapping taking values in a quotient space can be lifted through the quotient map. We prove two versions of said theorem for regular maps between Banach lattices. Our conditions mirror the classical ones, but additionally taking into account the order structure. In the first version, the Bounded Approximation Property is replaced by the Bounded Positive Approximation Property. In the second version, the property of being an \(L_1\)-predual is replaced by having Cartwright's property \((C)\): the connection stems from the fact that each of these conditions characterizes having a bidual which is injective (as a Banach space in the former case, and as a Banach lattice in the latter one).

We introduce the classes of operator \(p\)-compact mappings and completely right \(p\)-nuclear operators, which are natural extensions to the operator space framework of their corresponding Banach operator ideals. We relate these two classes, define natural operator space structures on them, and study several properties of these ideals. We show that the class of operator \(\infty\)-compact mappings in fact coincides with a notion already introduced by Webster in the nineties (in a very different language). This allows us to provide an operator space structure to Webster's class.

We define the frame potential for a Schauder frame on a finite dimensional Banach space as the square of the 2-summing norm of the frame operator. As is the case for frames for Hilbert spaces, we prove that the frame potential can be used to characterize finite unit norm tight frames (FUNTFs) for finite dimensional Banach spaces. We prove the existence of FUNTFs for a variety of spaces, and in particular that every \(n\)-dimensional complex Banach space with a 1-unconditional basis has a FUNTF of \(N\) vectors for every \(N \ge n\). However, many interesting results on FUNTFs and sums of rank-one projections for Hilbert spaces remain unknown for Banach spaces and we conclude the paper with multiple open questions.

We prove a version of the Ando–Choi–Effros lifting theorem respecting subspaces, which in turn relies on Oja's principle of local reflexivity respecting subspaces. To achieve this, we first develop a theory of pairs of \(M\)‐ideals. As a first consequence we get a version respecting subspaces of the Michael–Pełczyński extension theorem. Other applications are related to linear and Lipschitz bounded approximation properties (BAPs) for a pair consisting of a Banach space and a subspace. We show that in the separable case, the BAP for such a pair is equivalent to the simultaneous splitting of an associated pair of short exact sequences given by a construction of Lusky. We define a Lipschitz version of the BAP for pairs, and study its relationship to the (linear) BAP for pairs. The two properties are not equivalent in general, but they are when the pair has an additional Lipschitz‐lifting property in the style of Godefroy and Kalton. We also characterize, in the separable case, those pairs of a metric space and a subset whose corresponding pair of Lipschitz‐free spaces has the BAP.

Given an ideal \(\mathcal{A}\) of Lipschitz maps, we study the structure of the collection of all Lipschitz maps \(f : X \to Y\) between metric spaces \(X\) and \(Y\) satisfying the following property: not only can \(f\) be extended to every metric space containing \(X\), but the extension can be taken to be in the ideal \(\mathcal{A}\). We call such maps \(\mathcal{A}\)-extendible, and show that various properties can be inherited from the class \(\mathcal{A}\) to the class of \(\mathcal{A}\)-extendible maps. These include being an ideal, forming a Banach space and being a dual space. We also prove a characterization of these maps in terms of factorizations through absolute Lipschitz retracts.

In this paper, a new concept of weakly \(p\)-Cauchy sequences is introduced to study \(p\)-converging operators and Dunford–Pettis property of order \(p\). The notion of \(p\)–\((V)\) sets is defined to characterize the Dunford–Pettis property of order \(p\). We also introduce a complete locally convex topology by relatively weakly \(p\)-compact sets to quantify the Dunford–Pettis property of order \(p\).

We introduce the concepts of Pełczyński's property \((V)\) of order \(p\) and Pełczyński's property \((V^*)\) of order \(p\). It is proved that, for each \(1< p< \infty\), the James \(p\)-spaces \(J_p\) enjoy Pełczyński's property \((V^*)\) of order \(p\) and the James \(p^*\)-spaces \(J_{p^*}\) (where \(p^*\) denotes the conjugate number of \(p\)) enjoy Pełczyński's property \((V)\) of order \(p\). We prove that both \(L_1(\mu)\) (with \(\mu\) a finite positive measure) and \(\ell_1\) enjoy a quantitative version of Pełczyński's property \((V^*)\).

We develop a systematic approach to the study of ideals of Lipschitz maps from a metric space to a Banach space, inspired by classical theory on using Lipschitz tensor products to relate ideals of operator/tensor norms for Banach spaces. We study spaces of Lipschitz maps from a metric space to a dual Banach space that can be represented canonically as the dual of a Lipschitz tensor product endowed with a Lipschitz cross-norm, and we show that several known examples of ideals of Lipschitz maps (Lipschitz maps, Lipschitz \(p\)-summing maps, maps admitting Lipschitz factorization through a subset of an \(L_p\)-space) admit such a representation. We characterize when a space of Lipschitz maps from a metric space to a dual Banach space is in canonical duality with a Lipschitz cross-norm. Finally, we introduce a concept of operators which are approximable with respect to one of these ideals of Lipschitz maps, and we identify them in terms of tensor-product notions.

There are known results showing a canonical association between Lipschitz cross-norms (norms on the Lipschitz tensor product of a metric space and a Banach space) and ideals of Lipschitz maps from a metric space to a dual Banach space. We extend this association, relating Lipschitz cross-norms to ideals of Lipschitz maps taking values in general Banach spaces. To do that, we prove a Lipschitz version of the representation theorem for maximal operator ideals. As a consequence, we obtain linear characterizations of some ideals of (nonlinear) Lipschitz maps between metric spaces.

We introduce the operator space versions of the Chevet-Saphar tensor norms, and show that they share many properties with their Banach-space counterparts. Most importantly, these tensor norms are in operator space duality with the completely \(p\)-summing maps of G. Pisier. Our approach, based on the algebraic idea of a tensor contraction, complements previous results on duality for completely \(p\)-summing maps due to E. Effros and Z.-J. Ruan, and M. Junge. We also obtain an operator-space version of the Chevet-Persson-Saphar inequalities and give several applications of it, including a completely isomorphic characterization of quotients of subspaces of ultrapowers of the Schatten \(p\)-class.

Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product \(X \boxtimes E\) of a pointed metric space \(X\) and a Banach space \(E\) as a certain linear subspace of the algebraic dual of the space of \(E^*\)-valued functions on \(X\). We prove that \(X \boxtimes E\)is linearly isomorphic to the linear space of all finite-rank continuous linear operators from \((X^\#,\tau_p)\) to \(E\), where \(X^\#\) is the Lipschitz dual of \(X\) and \(\tau_p\) is the topology of pointwise convergence in \(X^\#\). To ensure the good behavior of a norm on \(X \boxtimes E\) with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on \(X \boxtimes E\) is defined. We show that the Lipschitz injective norm ε, the Lipschitz projective norm π and the Lipschitz p-nuclear norm \(d_p\) (\(1 \le p \le \infty\) ) are uniform dualizable Lipschitz cross-norms on \(X \boxtimes E\). In fact, ε is the least dualizable Lipschitz cross-norm and π is the greatest Lipschitz cross-norm on \(X \boxtimes E\) . Moreover, dualizable Lipschitz cross-norms on \(X \boxtimes E\) are characterized as those being between ε and π. In addition, the Lipschitz injective (projective) norm on \(X \boxtimes E\) can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product of the Lipschitz-free space over \(X\) and \(E\), but this identification does not hold for the Lipschitz 2-nuclear norm and the corresponding Banach-space tensor norm.

The notions of \(p\)-convexity and \(q\)-concavity are mostly known because of their importance as a tool in the study of isomorphic properties of Banach lattices, but they also play a role in several results involving linear maps between Banach spaces and Banach lattices. In this paper we introduce Lipschitz versions of these concepts, dealing with maps between metric spaces and Banach lattices, and start by proving nonlinear versions of two well-known factorization theorems through \(L_p\) spaces due to Maurey/Nikishin and Krivine. We also show that a Lipschitz map from a metric space into a Banach lattice is Lipschitz \(p\)-convex if and only if its linearization is \(p\)-convex. Furthermore, we elucidate why there is such a close relationship between the linear and nonlinear concepts by proving characterizations of Lipschitz \(p\)-convex and Lipschitz \(q\)-concave maps in terms of factorizations through \(p\)-convex and \(q\)-concave Banach lattices, respectively, in the spirit of the work of Raynaud and Tradacete.

The famous Mazur Rotation Problem asks whether any separable transitive Banach space (that is, a Banach space where any point on the unit sphere can be mapped into any other point on the unit sphere by a surjective isometry) is necessarily isometric to a Hilbert space. In spite of enormous progress since the 1930’s, the problem remains open. In this paper we investigate related non-commutative phenomena. We show that the only completely uniquely maximal (or matrix convex transitive) operator space is a one-dimensional one. Relaxing the conditions somewhat, we show that any matrix-level convex transitive finite dimensional space has to be completely isometric to Pisier’s space OH, of corresponding dimension. Finally, we equip \(\ell_2\) with an operator space structure which is (i) completely almost transitive, and (ii) homogeneous, but not 1-homogeneous.

There are well-known relationships between compressed sensing and the geometry of the finite-dimensional \(\ell_p\) spaces. A result of Kashin and Temlyakov can be described as a characterization of the stability of the recovery of sparse vectors via \(\ell_1\)-minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional \(\ell_1\) and \(\ell_2\) spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich proves an analogous relationship even for \(\ell_p\) spaces with \(p< 1\). In this paper we prove what we call matrix or noncommutative versions of these results: we characterize the stability of low-rank matrix recovery via Schatten \(p\)-(quasi-)norm minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional Schatten \(p\)-spaces.

The Euclidean distortion of a metric space, a measure of how well the metric space can be embedded into a Hilbert space, is currently an active interdisciplinary research topic. We study the corresponding notion for mappings instead of spaces, which is that of Lipschitz factorization through subsets of Hilbert space. The main theorems are two characterizations of when a mapping admits such a factorization, both of them inspired by results dealing with linear factorizations through Hilbert space. The first is a nonlinear version of a classical theorem of Kwapień in terms of “dominated” sequences of vectors, whereas the second is a duality result by means of a tensor-product approach.

Several important results for \(p\)-summing operators, such as Pietsch’s composition formula and Grothendieck’s theorem, share the following form: there is an operator \(T\) such that \(S \circ T\) is \(p\)-summing whenever \(S\) is \(q\)-summing. Such operators were called \((q,p)\)-mixing by Pietsch, who studied them systematically. In the operator space setting, G. Pisier’s completely \(p\)-summing maps correspond to the \(p\)-summing operators between Banach spaces. A natural modification of the definition yields the notion of completely \((q,p)\)-mixing maps, already introduced by K. L. Yew, which is the subject of this paper. Some basic properties of these maps are proved, as well as a couple of characterizations. A generalization of Yew’s operator space version of the Extrapolation theorem is obtained, via an interpolation-style theorem relating different completely \((q,p)\)-mixing norms. Finally, some composition theorems for completely \(p\)-summing maps are proved.

Several useful results in the theory of \(p\)-summing operators, such as Pietsch’s composition theorem and Grothendieck’s theorem, share a common form: for certain values \(q\) and \(p\), there is an operator such that whenever it is followed by a \(q\)-summing operator, the composition is \(p\)-summing. This is precisely the concept of \((q,p)\)-mixing operators, defined and studied by A. Pietsch. On the other hand, J. Farmer and W. B. Johnson recently introduced the notion of a Lipschitz \(p\)-summing operator, a nonlinear generalization of \(p\)-summing operators. In this paper, a corresponding nonlinear concept of Lipschitz \((q,p)\)-mixing operators is introduced, and several characterizations of it are proved. An interpolation-style theorem relating different Lipschitz \((q,p)\)-mixing constants is obtained, and it is used to show reversed inequalities between Lipschitz \(p\)-summing norms.

Building upon the ideas of R. Arens and J. Eells we introduce the concept of spaces of Banach-space-valued molecules, whose duals can be naturally identified with spaces of operators between a metric space and a Banach space. On these spaces we define analogues of the tensor norms of Chevet and Saphar, whose duals are spaces of Lipschitz \(p\)-summing operators. In particular, we identify the dual of the space of Lipschitz \(p\)-summing operators from a finite metric space to a Banach space — answering a question of J. Farmer and W.B. Johnson — and use it to give a new characterization of the non-linear concept of Lipschitz \(p\)-summing operator between metric spaces in terms of linear operators between certain Banach spaces. More generally, we define analogues of the norms of J.T. Lapresté, whose duals are analogues of A. Pietschʼs \((p,r,s)\)-summing operators. As a special case, we get a Lipschitz version of \((q,p)\)-dominated operators.

Dynamical sampling is a paradigm in signal processing based on the following desirable spatio-temporal trade-off: instead of recovering a signal by measuring it at many different locations at the same time (requiring multiple sensors), we would like to be able to recover it by measuring it only at one location but at many different times (requiring only one sensor). In the Hilbert space setting, frames (a generalization of orthonormal bases) are a well-established method of doing signal recovery. The dynamical sampling question can then be shown to translate to: for which bounded linear operators on a Hilbert space does there exist an orbit which is a frame? This question has by now been fully answered, together with other closely related ones. Using the notion of Schauder frames for Banach spaces, we explore several versions of dynamical sampling questions in this context and prove various results reminiscent of the classical ones. While some of these results are less explicit, they are easier to understand conceptually.

We further the study a notion of completely bounded Lipschitz functions defined on the open unit ball of an operator space, already considered in [B.M. Braga and J.A. Chávez-Domí:nguez, On the small scale nonlinear theory of operator spaces. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1893–1914]. We endow the set of these functions with an operator space structure, and in the scalar-valued case we identify an operator space predual for it which is a noncommutative version of the standard Lipschitz-free space. In particular, our predual is a free operator space in the sense that it satisfies a linearization property for vector-valued completely bounded Lipschitz functions. Additionally, we study some lifting properties in the style of Godefroy and Kalton. Since our functions can only be defined on balls and not on a full operator space, many of the classical techniques cannot be directly adapted.

We study operator space versions of the asymptotic tensor powers of Banach spaces of Aubrun and Müller-Hermes (arXiv:2110.12828 [math.FA]). While some of the aspects are straight translations of the Banach space situation (e.g. the relation with nuclear operators), others are completely different. For example, in the classical setting the tensor radius of an \(n\)-dimensional Banach space is between \(\sqrt{n}\) and \(n\), with the upper bound being achieved exactly at Hilbertian spaces. In the noncommutative setting, the tensor radius of an \(n\)-dimensional operator space can be as low as 1, regardless of the dimension, and for Pisier's Hilbertian operator space OH the radius is equal to \(\sqrt{n}\). Due to this difference, the averaging techniques of Aubrun and Müller-Hermes cannot yield sharp results in the noncommutative setting, and therefore new ideas are needed.

If \((E_n)_{n=1}^\infty\) is an increasing sequence of finite-dimensional subspaces of a Banach space \(X\) whose union is dense in \(X\), and \(c\big((E_n)_{n=1}^\infty\big)\) denotes the space of convergent sequences in \(X\) such that the \(n\)-th term belongs to \(E_n\), the map \(q :c\big((E_n)_{n=1}^\infty\big) \to X \) defined by taking the limit of a sequence is in fact a quotient map. It is well-known that the existence of a bounded linear lifting for this quotient map characterizes the Bounded Approximation Property for \(X\): this construction has been exploited by Lusky in the 80's, and Johnson and Oikhberg in the early 2000's. More recently, this construction has also played a recurrent role in the study of approximation properties for Lipschitz-free spaces, for example in works by Godefroy, Godefroy and Ozawa, Borel-Mathurin, and the author. In this project we show that a similar construction can be used to characterize the Completely Bounded Approximation Property for operator spaces. Most of the techniques we use were developed by Effros and Ruan [Mapping spaces and liftings for operator spaces, Proc. London Math. Soc. (3) 69 (1994), no. 1, 171–197]. As is often the case in the noncommutative setting, we need to assume local reflexivity.

A frame in a finite-dimensional Hilbert space is an "overcomplete basis", that is, a redundant coordinate system. This redundancy is important in applications, where measurements are noisy and can even be lost. Frames consisting of vectors of norm one and additionally satisfying a technical condition called tightness (also known as FUNTFs) are particularly desirable, since they minimize both the mean squared error due to noise and the reconstruction error due to the loss of a single coefficient. One way of constructing such frames, which appears often in applications, is by taking the orbit of a single vector under the action of a group acting by unitaries on the Hilbert space. More generally, Vale and Waldron [Tight frames and their symmetries, Constr. Approx. 21 (2005), no. 1,83–112], [The construction of \(G\)-invariant finite tight frames, J. Fourier Anal. Appl. 22 (2016), no.5, 1097–1120] have studied the construction of tight frames by taking unions of such orbits. Recently the notion of a FUNTF has been extended to general finite-dimensional Banach spaces by the author and collaborators [Frame potential for finite-dimensional Banach spaces, Linear Algebra Appl. 578 (2019), 1–26]. The main goal of this project is to characterize when one can obtain FUNTFs for finite-dimensional Banach spaces by taking orbits of vectors under the action of a group which is acting by linear isometries on the Banach space. The results look similar to the classical Hilbert-space case, in the sense that the "mass" of the vectors whose orbits are being taken has to be distributed among the irreducible subspaces in a way proportional to their dimension.