I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at C*-algebras based on my interests. However specialists in the field seem to be rather rare and in expanding my search it seems that Banach Algebra's are closely related but I'm not entirely sure on the distinction (other than C*-algebras seem to be a specific form of Banach's). Can someone tell me the difference?
2026-03-28 17:42:01.1774719721
What's the difference between a Banach Algebra and a C*-algebra?
456 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ANALYSIS
- Analytical solution of a nonlinear ordinary differential equation
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Show that $d:\mathbb{C}\times\mathbb{C}\rightarrow[0,\infty[$ is a metric on $\mathbb{C}$.
- conformal mapping and rational function
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Elementary question on continuity and locally square integrability of a function
- Proving smoothness for a sequence of functions.
- How to prove that $E_P(\frac{dQ}{dP}|\mathcal{G})$ is not equal to $0$
- Integral of ratio of polynomial
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 BANACH-ALGEBRAS
- Bijection between $\Delta(A)$ and $\mathrm{Max}(A)$
- To find an element in $A$ which is invertible in $B$ but not in $A$.
- Let $\varphi: A \to \mathbb C$ be a non-zero homomorphism. How can we extend it to an homomorphism $\psi: \overline A \to \mathbb C$?
- Prove that the set of invertible elements in a Banach algebra is open
- Separability of differentiable functions
- An injective continuous map between two compact Hausdorff spaces.
- Banach algebra of functions under composition
- Double limit of a net
- Can we characterise $X$ being separable in terms of $C(X, \mathbb R)$?
- Unit ball of the adjoint space of a separable Banach space is second-countable in the weak* topology.
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?
It is true: $C^*$-algebras are a special class of Banach algebras. But they have a very rich structure as they appear as selfadjoint algebras of operators on a complex Hilbert space. The theory of $C^*$-algebras is very developt. On the other hand, general Banach algebras are a much wider class of algebras and working on them one has to assume some additional conditions. Actually any Banach space can be made in a Banach algebra by a trivial multiplication, or by a multiplication given by $x.y=\xi(x)y$, where $\xi$ is a fixed bounded linear functional on the space... In any case, for working on $C^*$-algebras, one has to know the general theory of Banach algebras and, on the other hand, working in general Banach algebras, one has to be familiar with $C^*$-algebras as they are a very important class of Banach algebras.