I'm currently enrolled in an operator spaces course and I'm finding it difficult to understand why we study them in the first place. Functional analysis is motivated well enough for me and even though I don't have a firm grasp on them, I guess I can see why $C^{\ast}$-algebras are studied as well (purely from a quantum mechanics point of view). However I fail to see what the motivation is for studying operator spaces or why they're useful/important. What is their motivation? What led people to be interested in them and what is so special about completely bounded/positive maps that we study them?
2026-02-23 06:33:16.1771828396
Why study operator spaces?
1.8k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in FUNCTIONAL-ANALYSIS
- On sufficient condition for pre-compactness "in measure"(i.e. in Young measure space)
- Why is necessary ask $F$ to be infinite in order to obtain: $ f(v)=0$ for all $ f\in V^* \implies v=0 $
- Prove or disprove the following inequality
- Unbounded linear operator, projection from graph not open
- $\| (I-T)^{-1}|_{\ker(I-T)^\perp} \| \geq 1$ for all compact operator $T$ in an infinite dimensional Hilbert space
- Elementary question on continuity and locally square integrability of a function
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- Exercise 1.105 of Megginson's "An Introduction to Banach Space Theory"
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- If $A$ generates the $C_0$-semigroup $\{T_t;t\ge0\}$, then $Au=f \Rightarrow u=-\int_0^\infty T_t f dt$?
Related Questions in OPERATOR-ALGEBRAS
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- hyponormal operators
- Cuntz-Krieger algebra as crossed product
- Identifying $C(X\times X)$ with $C(X)\otimes C(X)$
- If $A\in\mathcal{L}(E)$, why $\lim\limits_{n\to+\infty}\|A^n\|^{1/n}$ always exists?
- Given two projections $p,q$ in a C$^{*}$-algebra $E$, find all irreducible representations of $C^{*}(p,q)$
- projective and Haagerup tensor norms
- AF-algebras and K-theory
- How to show range of a projection is an eigenspace.
- Is $\left\lVert f_U-f_V\right\rVert_{op}\leq \left\lVert U-V\right\rVert_2$ where $f_U = A\mapsto UAU^*$?
Related Questions in OPERATOR-SPACES
- Hahn Banach extension type theorem
- Hilbert space as an operator space.
- I have question about the operator space theory
- The spectral transfinite open spaces with quintic characteristics of second kind
- kernel of Haagerup tensor product of maps
- Projective tensor product of operator spaces
- Completion of operator space
- completely positivity on operator system category implies completely contractivity on operator space category?
- max-min nuclear operator system
- Understanding the definition of completely bounded bilinear map
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
In short the list of reasons is:
The modern trend in math today is to develop non-commutative analogues of well known theories. Vaguely speaking operator spaces are normed spaces over "non-commutative" scalars, in fact over matricies.
There were several long standing problems, that were solved via methods of operator space theory. As soon as a problem is embedded in its natural environment the solution comes in a natural way. For example, thanks to D. P. Blecher, we have a criterion for a Banach algebra $A$ to be isomorphic as an algebra to a closed subalgebra of $\mathcal{B}(H)$: multiplication in the Banach algebra $A$ must be completely bounded for some embedding (as a normed space) of $A$ into some space of bounded operators.
Mathematicians not only prove theorems but also guess good definitions. Whether a definition is good or not will become evident only after its usage in theorems. It turns out that some notions are better to define withing the scope of operator space theory. For example a criterion of amenability of the Fourier algebra $A(G)$ for a locally compact group $G$ was very unnatural. But if we consider not a simple amenability but an operator amenability then we get a nice characterization in the sense of B. Johnson: Fourier algebra is operator ameanable iff its group is ameanable.
Operator spaces possess some unexpected interesting properties. For example, in this theory we have a tensor product which both injective and projective and what is more it is not commutative! Also, thanks to W. F. Steinspring, we have quite explicit description of maps between operator spaces. For classical normed spaces this problem is hopeless.
The most I've written here is a a copy-paste from the book Quantum Functional Analysis: Non-coordinate approach by A. Ya. Helemskii.